Collimated outflows of rapidly rotating young stellar objects - Wind equation, GSS equation and collimation

The magnetic field structure and the collimated outflow of rapidly rotating Young Stellar Objects (YSOs) are calculated from the stellar source to the asymptotic region. The calculations are based on ideal MHD and the further simplifying assumptions of stationarity and axisymmetry. The star-disk-jet system can be completely defined by the Grad-Shafranov (GSS) equation, des cribing the structure of the magnetospheres, and the wind equation, which a re given by equilibrium perpendicular and parallel to the field lines. Both equations must be solved simultaneously to obtain a self-consistent soluti on. General solutions of the Grad-Skafranov equation are not yet available. Here we discuss an analytical model for the magnetic flux surfaces which i s a solution for small and large radii. This model assumes a stellar dipole field and a gap between the star and a disk at a distance of a few stellar radii. Due to the features of the disk no field can penetrate the disk and the resulting opening of the field lines close to the polar cap is obtaine d as a computational result. In addition, our model guarantees the collimat ion of the outflow into a cylindrical shape at asymptotic jet radii of seve ral thousand stellar radii. This model for collimated outflows reproduces a ll essential properties of magnetospheres for rapidly rotating stars. The r esult is used as input to the wind equation. This problem is completely int egrable, determined by five constants of motion: the total energy E, the to tal angular momentum L, the total mass flux eta along a flux surface, and t he total entropy S in a flux surface, together with the: rotation Omega(F) of the field lines. For adiabatic plasma flows, this problem is algebraic a nd can easily be solved. Pressure is neglected in our computation. The theory of axisymmetric magnetospheres around rapidly rotating stellar s ources is outlined including electric fields due to the rapid rotation. Gra vity of the central object is consistently built into this theory. Due to t he injection of plasma either from the stellar surface or by interaction wi th a surrounding disk, these magnetospheres are neither vacuum solutions, n or force-free. A consistent wind theory is developed which contains the New tonian theory as a classical limit. Current-carrying plasma flows will lead to a collimation of the magnetosphe ric structure into a cylindrical shape. Particular solutions are discussed for the asymptotic collimation. We show that the asymptotic structure is es sentially determined by electric forces in the pinch equation, and not by c entrifugal and pressure forces.